Question

lesson: modeling the product of polynomials (area model and multiplication table)assessmentguided notes1. what is the product of $(0.5x + 3)$ and $(x - 2)$?a. $0.5x^2 + 3x - 6$b. $0.5x^2 - 1.5x - 6$c. $0.5x^2 + 0.5x - 6$d. $x^2 + 2x - 6$2. how would you describe the interaction of negative coefficients in the area model?a. they result in negative areasb. they are ignoredc. they are addedd. they are multiplied by zero3. what is the final simplified form of the product $(x - 4)(x + 5)$?a. $x^2 - 20$b. $x^2 + 5x - 4x$c. $x^2 - x - 20$d. $x^2 + x - 20$

Explanation:

Step1: Multiply first polynomials

Use distributive property:

$$\begin{align*} (0.5x + 3)(x - 2) &= 0.5x(x - 2) + 3(x - 2)\\ &= 0.5x^2 - x + 3x - 6\\ &= 0.5x^2 + 2x - 6 \end{align*}$$

Step2: Analyze negative coefficients

In area models, negative coefficients correspond to negative area regions that represent subtractions of area, so they result in negative areas.

Step3: Multiply third polynomials

Use distributive property (FOIL method):

$$\begin{align*} (x - 4)(x + 5) &= x^2 + 5x - 4x - 20\\ &= x^2 + x - 20 \end{align*}$$

Answer:

1. c. $0.5x^2 + 2x - 6$
2. a. They result in negative areas
3. d. $x^2 + x - 20$